Calculus of variations centered around exponentially harmonic maps

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K17546
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Tokyo University of Science
• Principal Investigator: Omori Toshiaki 東京理科大学, 理工学部数学科, 助教 (20638225)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: 指数調和写像 / 調和写像 / 離散曲面 / グラフスペクトル

Outline of Final Research Achievements

Several existence theorems for exponentially harmonic maps are obtained. More precisely, an existence theorem for exponentially harmonic maps in the case that the source manifold is noncompact, and a Liouville-type theorem is also obtained for exponentially harmonic maps with bounded energy whose target has nonpositive curvature. The existence of a time-global solution to a time evolutional equation for exponentially harmonic maps into nonpositively curved manifolds has been proved. Moreover, the existence of a kind of equivariant exponentially harmonic maps between spheres has been proved under without any conditions.
Also, from a viewpoint of the area of material science, some realizations of graphs in the Euclidean space, a discrete surface theory for them, and a continuous limit of their subdivisions are studied. Spectral problems of the Goldberg-Coxeter subdivisions for 3- and 4-valent finite graphs are also studied.

Free Research Field

幾何解析および離散幾何解析

Academic Significance and Societal Importance of the Research Achievements

本研究は，これまでのリーマン多様体間の調和写像の存在理論に対して，指数調和写像を用いるという新しい手法により，統一的な理論の理解を与えたという点において価値がある。
また，連続曲面の離散化でもなく，必ずしも面の存在を仮定しない次数3の空間グラフに対して離 散曲面論を構築した。この研究は，既存の曲面論の枠にはまらない非多面体曲面を対象にする という点で新しいものである。